The problem I'm struggling with is to solve the PDE,
$$ (xz-y) \frac{\partial{z}}{\partial{x}} + (yz-x) \frac{\partial{z}}{\partial{y}}=1-z^2 $$
with $x$ and $y$ being independent variables.
What I Tried
The only way I know to solve this first-order quasilinear PDE is using the characteristic system of equations: $$ \frac{\partial{x}}{xz-y} = \frac{\partial{y}}{yz-x} = \frac{\partial{z}}{1-z^2}$$ The two equations don't seem to be easy to solve directly. I considered finding coeffecients $l(x)$, $m(y)$, and $n(z)$ that satisfy $l(xz-y) + m(yz-x) + n(1-z^2)=0$ but I could not succeed.
Any help will be appreciated.